# Lifting Etale Morphisms

I am trying to find a reference for the following theorem:

Let $R$ be a complete DVR, and let $Y$ be a scheme projective and flat over $R$. Suppose that $X_0 \longrightarrow Y_0$ is a finite etale morphism, where $Y_0$ is the fiber of $Y$ over the unique closed point of Spec $R$. Then there exists a scheme $X$, finite etale over $Y$, together with a closed immersion $X_0 \longrightarrow X$ such that $X_0 = X \times_Y Y_0$.

In a special case, this allows us to lift etale morphisms in characteristic $p$ to etale morphisms in characteristic zero, and can thus be important in studying the etale fundamental group of varieties in finite characteristic.

Note: While any reference would be appreciated, what I am really interested in is a reference that I can give for attribution purposes. My advisor thinks that this result was proved by Deligne about 40 years ago, but I would like something a little more solid.

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The study of etale pi_1 in char > 0 was one of the first huge achievements of the theory of schemes (since the above innocuous-sounding result needs everything, including formal GAGA, especially when assuming just properness, not projectivity or flatness). So check out the old FGA exposes to see if it may even be stated there (likely without proof, given in sga1 later). –  BCnrd May 5 '10 at 22:29
Thanks for the reference and the information. [Unless my advisor and I are both horrendously mistaken, the specific case I stated (assuming projectivity and flatness) does not require formal GAGA.] –  Charles Staats May 5 '10 at 23:12
OK, how do you make the coherent sheave of algebras on $Y$ corresponding to $X$ without using formal GAGA? Or rather, how do you explot projectivity and flatness to make a simpler proof? (Could be easy, I haven't tried in this special case.) –  BCnrd May 5 '10 at 23:36
Let Y_i be the product of Y with R/(t^{i+1}). Inductively lift X_i -> Y_i to finite etale X_{i+1} -> Y_{i+1}. Let L be an ample sheaf on Y, and pull this back to sheaves on Y_i and then X_i, which remain ample by a cohomology argument. If M_i is the ample sheaf on X_i, let A_{i,d} be the global sections of the dth tensor power of M_i. Then the direct sum A_i of the A_{i,d} is a graded ring whose Proj is X_i. Define a graded ring A by letting A_d be the inverse limit of the A_{i,d}. Then Proj A is the desired X. –  Charles Staats May 6 '10 at 3:41
Flatness comes into play in showing that X -> Y is actually etale, as well as showing that first cohomology of M_i^{\otimes d} vanishes for d sufficiently large. (This is important in some of the details I have omitted.) –  Charles Staats May 6 '10 at 3:44

SGA 1, IX, 1.10: Let $Y$ be a scheme proper over a complete local noetherian ring $R$, and let $Y_0$ be the closed fibre of $Y/R$. Then the functor $X\mapsto X_0$ from finite etale coverings of $Y$ to finite etale coverings of $Y_0$ is an equivalence of categories.